Question Number 150376 by cherokeesay last updated on 11/Aug/21
Commented by liberty last updated on 12/Aug/21
$$\left(\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{r}\right)^{\mathrm{2}} =\left(\mathrm{4}\sqrt{\mathrm{3}}−\mathrm{r}\right)^{\mathrm{2}} +\left(\mathrm{2}\sqrt{\mathrm{3}}−\mathrm{r}\right)^{\mathrm{2}} \\ $$$$\mathrm{let}\:\mathrm{2}\sqrt{\mathrm{3}}\:=\:{a} \\ $$$$\Rightarrow\left({a}+{r}\right)^{\mathrm{2}} −\left({a}−{r}\right)^{\mathrm{2}} =\left(\mathrm{2}{a}−{r}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{2}{a}\left(\mathrm{2}{r}\right)=\mathrm{4}{a}^{\mathrm{2}} −\mathrm{4}{ar}+{r}^{\mathrm{2}} \\ $$$$\Rightarrow{r}^{\mathrm{2}} −\mathrm{8}{ar}+\mathrm{4}{a}^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow{r}=\frac{\mathrm{8}{a}\pm\sqrt{\mathrm{64}{a}^{\mathrm{2}} −\mathrm{16}{a}^{\mathrm{2}} }}{\mathrm{2}} \\ $$$$\Rightarrow{r}=\mathrm{4}{a}\pm\mathrm{2}{a}\sqrt{\mathrm{3}}\:=\:\mathrm{8}\sqrt{\mathrm{3}}\:\pm\mathrm{12} \\ $$$$\mathrm{r}=\mathrm{8}\sqrt{\mathrm{3}}+\mathrm{12}\:\leftleftarrows\:\mathrm{not}\:\mathrm{possible} \\ $$$$\mathrm{r}=\mathrm{8}\sqrt{\mathrm{3}}−\mathrm{12}\:\leftleftarrows\:\mathrm{possible} \\ $$$$ \\ $$
Commented by cherokeesay last updated on 12/Aug/21
$${thank}\:{you}\:! \\ $$
Answered by lyubita last updated on 11/Aug/21
$$\mathrm{8}\sqrt{\mathrm{3}}\:−\:\mathrm{12} \\ $$
Commented by cherokeesay last updated on 11/Aug/21
$${thanks}\:! \\ $$